Electron energy-loss spectroscopy (EELS) is shown to be an excellent source of information both on photonic crystal bands and on radiation modes of complex nanostructures. Good agreement is reported between measurements and parameter-free calculations of EELS in porous alumina films, where Cherenkov radiation is scattered by the pores to yield a strong 8.3-eV (7-eV) feature for 120-keV (200-keV) electrons. The latter is related to the bands of two-dimensional photonic crystals formed by air cylinders in an alumina matrix with similar near-range ordering. Finally, the band structure is proved to be directly mapped by angle-resolved EELS.